A Note On Elliptic Plane Curves With Fixed j-Invariant

TL;DR

This paper derives a formula for counting elliptic plane curves with a fixed j-invariant passing through a specific number of points, extending classical enumerative geometry results for rational curves.

Contribution

It provides a new explicit count for elliptic plane curves with fixed j-invariant, relating it to known counts of rational curves and binomial coefficients.

Findings

Number of elliptic curves with fixed j-invariant: {d-1 extbackslash choose 2} * N_d

Established a direct relation between elliptic and rational curve counts

Extended enumerative geometry to include fixed j-invariant conditions

Abstract

Let N_d be the number of degree d, nodal, rational plane curves through 3d-1 points in the complex projective plane. The number of degree d>=3, nodal, elliptic plane curves with a fixed (general) j-invariant through 3d-1 points is found to be {d-1 \choose 2}*N_d.